Abstract

We use three-dimensional (3D) smoothed particle hydrodynamics (SPH) calculations with higher resolution, as well as with more realistic viscosity and sound-speed prescriptions than previous work to examine the eccentric instability which underlies the superhump phenomenon in semidetached binaries. We illustrate the importance of the two-armed spiral mode in the generation of superhumps. Differential motions in the fluid disc cause converging flows which lead to strong spiral shocks once each superhump cycle. The dissipation associated with these shocks powers the superhump. We compare two-dimensional (2D) and 3D results, and conclude that 3D simulations are necessary to faithfully simulate the disc dynamics. We ran our simulations for unprecedented durations, so that an eccentric equilibrium is established except at high mass ratios where the growth rate of the instability is very low.

We collate the observed data on superhumps. Our improved simulations give a closer match to the observed relationship between superhump period excess and binary mass ratio than previous numerical work. The observed black hole X-ray transient superhumpers appear to have systematically lower disc precession rates than the cataclysmic variables. This could be due to higher disc temperatures and thicknesses. No high-resolution 3D disc with mass ratio q > 0.24 developed superhumps, in agreement with analytical expectations.

The modulation in total viscous dissipation on the superhump period is overwhelmingly from the region of the disc within the 3:1 resonance radius R3:1. The precession rates of our high-resolution 3D discs match the single particle dynamical precession rate at 0.87 R3:1. As the eccentric instability develops, the viscous torques are enhanced, and the disc consequently adjusts to a new equilibrium state, as suggested in the thermal–tidal instability model. We quantify this enhancement in the viscosity, which is 10 per cent for q = 0.08. The disc motions can be described as superpositions of the S(k, l) modes, and the disc executes complex standing wave dynamics which repeat in the inertial frame on the disc precession period. We characterize the eccentricity distributions in our accretion discs, and show that the entire body of the disc partakes in the eccentricity.